Proportional Relationships - ISEE Middle Level: Quantitative Reasoning

$25%$. Setting up $18/72 = p/100$ and solving gives $p = (18/72) imes 100 = 25$.

$70$ miles. The speed is $42/3 = 14$ mph, so in 5 hours the distance is $5 imes 14 = 70$ miles.

$\$25$. The unit price is $10/6 = 5/3$, so for 15 items the cost is $15 imes (5/3) = 25$.

$\$\frac{4}{3}$ per item. Dividing total cost by number of items gives the rate of $24/18$, which simplifies to $4/3$ per item.

Yes, because $5\cdot24=6\cdot20$. The cross-products are equal at 120, confirming the ratios are proportional.

A) $\frac{2}{3}$ and $\frac{8}{12}$. Simplifying $8/12$ to $2/3$ shows equivalence to the first ratio, while $8/10$ simplifies to $4/5$, which differs.

$x=15$. Rearranging $y = kx$ to $x = y/k$ and substituting gives $x = 25 / (5/3) = 15$.

$y=15$. Substituting into the equation yields $y = (3/4) imes 20 = 15$.

$k=\frac{5}{2}$. Dividing $y$ by $x$ gives the constant $k=15/6$, which simplifies to $5/2$.

$x=16$. Cross-multiplying results in $3x = 48$, so dividing by 3 provides $x=16$.

$x=15$. Cross-multiplying yields $180 = 12x$, and dividing by 12 gives $x=15$.

$x=4$. Cross-multiplying gives $84 = 21x$, so dividing by 21 solves for $x=4$.

$x=4$. Cross-multiplying the proportions $15x = 60$ and dividing by 15 yields $x=4$.

$\frac{\text{part}}{\text{whole}}=\frac{\text{percent}}{100}$. The percent proportion sets the ratio of part to whole equal to the percent over 100 to solve for unknowns.

$y=kx$. This equation captures the direct variation where $y$ is always a constant multiple $k$ of $x$.

They are equal: $

\frac{a}{b}=\frac{c}{d}$ with $b\ne^0$ and $d\ne^0$. Two ratios are proportional when they express the same relationship, making their fractions equivalent provided the denominators are not zero.

$x=\frac{bc}{a}$ (with $a\ne^0$). Cross-multiplication equates $a x = b c$, so isolating $x$ gives the product of $b$ and $c$ divided by $a$.

$x=\frac{ad}{b}$ (with $b\ne^0$). Solving for the missing value uses cross-multiplication to equate products, yielding $x$ as the product of $a$ and $d$ divided by $b$.

$\frac{a}{b}$ per $1$ (the value when the second quantity is $1$). The unit rate simplifies the ratio to express the value of the first quantity when the second is exactly 1.

$(0,0)$. The graph of $y=kx$ is a straight line through the origin, so it always passes through $(0,0)$ regardless of $k$.

$y=kx$. A proportional relationship is modeled by this linear equation where $y$ varies directly with $x$ through the constant $k$.

$k=\frac{y}{x}$ for $x\ne^0$. The constant of proportionality $k$ represents the fixed ratio of $y$ to $x$ in a direct proportion, defined as their quotient when $x$ is not zero.

Proportional if and only if $ad=bc$ (with $b\ne^0$ and $d\ne^0$). The cross-products test checks equality by verifying if the product of the numerator of one ratio and the denominator of the other equals the reverse, excluding zero denominators.

$y=35$. The constant $k=14/8=7/4$, so for $x=20$, $y=20 imes (7/4)=35$.